In formal language theory, a context-free language (CFL), also called a Chomsky type-2 language, is a language generated by a context-free grammar (CFG).
Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars.
Background
[edit]Context-free grammar
[edit]Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language.
Automata
[edit]The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
Examples
[edit]An example context-free language is , the language of all non-empty even-length strings, the entire first halves of which are a's, and the entire second halves of which are b's. L is generated by the grammar . This language is not regular. It is accepted by the pushdown automaton where is defined as follows:[note 1]
Unambiguous CFLs are a proper subset of all CFLs: there are inherently ambiguous CFLs. An example of an inherently ambiguous CFL is the union of with . This set is context-free, since the union of two context-free languages is always context-free. But there is no way to unambiguously parse strings in the (non-context-free) subset which is the intersection of these two languages.[1]
Dyck language
[edit]The language of all properly matched parentheses is generated by the grammar .
Properties
[edit]Context-free parsing
[edit]The context-free nature of the language makes it simple to parse with a pushdown automaton.
Determining an instance of the membership problem; i.e. given a string , determine whether where is the language generated by a given grammar ; is also known as recognition. Context-free recognition for Chomsky normal form grammars was shown by Leslie G. Valiant to be reducible to boolean matrix multiplication, thus inheriting its complexity upper bound of O(n2.3728596).[2][note 2] Conversely, Lillian Lee has shown O(n3−ε) boolean matrix multiplication to be reducible to O(n3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.[3]
Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string. The process of producing this tree is called parsing. Known parsers have a time complexity that is cubic in the size of the string that is parsed.
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the CYK algorithm and Earley's Algorithm.
A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and can be parsed by a LR(k) parser.[4]
See also parsing expression grammar as an alternative approach to grammar and parser.
Closure properties
[edit]The class of context-free languages is closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:
- the union of L and P[5]
- the reversal of L[6]
- the concatenation of L and P[5]
- the Kleene star of L[5]
- the image of L under a homomorphism [7]
- the image of L under an inverse homomorphism [8]
- the circular shift of L (the language )[9]
- the prefix closure of L (the set of all prefixes of strings from L)[10]
- the quotient L/R of L by a regular language R[11]
Nonclosure under intersection, complement, and difference
[edit]The context-free languages are not closed under intersection. This can be seen by taking the languages and , which are both context-free.[note 3] Their intersection is , which can be shown to be non-context-free by the pumping lemma for context-free languages. As a consequence, context-free languages cannot be closed under complementation, as for any languages A and B, their intersection can be expressed by union and complement: . In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: .[12]
However, if L is a context-free language and D is a regular language then both their intersection and their difference are context-free languages.[13]
Decidability
[edit]In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar.
The following problems are undecidable for arbitrarily given context-free grammars A and B:
- Equivalence: is ?[14]
- Disjointness: is ?[15] However, the intersection of a context-free language and a regular language is context-free,[16][17] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
- Containment: is ?[18] Again, the variant of the problem where B is a regular grammar is decidable,[citation needed] while that where A is regular is generally not.[19]
- Universality: is ?[20]
- Regularity: is a regular language?[21]
- Ambiguity: is every grammar for ambiguous?[22]
The following problems are decidable for arbitrary context-free languages:
- Emptiness: Given a context-free grammar A, is ?[23]
- Finiteness: Given a context-free grammar A, is finite?[24]
- Membership: Given a context-free grammar G, and a word , does ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.
According to Hopcroft, Motwani, Ullman (2003),[25] many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of Bar-Hillel, Perles, and Shamir[26]
Languages that are not context-free
[edit]The set is a context-sensitive language, but there does not exist a context-free grammar generating this language.[27] So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages[26] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[28]
Notes
[edit]- ^ meaning of 's arguments and results:
- ^ In Valiant's paper, O(n2.81) was the then-best known upper bound. See Matrix multiplication#Computational complexity for bound improvements since then.
- ^ A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: S → Sc | aTb | ε; T → aTb | ε. The grammar for B is analogous.
References
[edit]- ^ Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
- ^ Valiant, Leslie G. (April 1975). "General context-free recognition in less than cubic time" (PDF). Journal of Computer and System Sciences. 10 (2): 308–315. doi:10.1016/s0022-0000(75)80046-8.
- ^ Lee, Lillian (January 2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication" (PDF). J ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241.505242. S2CID 1243491. Archived (PDF) from the original on 2003-04-27.
- ^ Knuth, D. E. (July 1965). "On the translation of languages from left to right". Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2.
- ^ a b c Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.
- ^ Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.
- ^ Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.
- ^ Hopcroft & Ullman 1979, p. 132, Theorem 6.3.
- ^ Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.
- ^ Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.
- ^ Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.
- ^ Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3 (4): 372–375. doi:10.1016/s0019-9958(60)90965-7. Archived (PDF) from the original on 2018-11-26.
- ^ Beigel, Richard; Gasarch, William. "A Proof that if L = L1 ∩ L2 where L1 is CFL and L2 is Regular then L is Context Free Which Does Not use PDA's" (PDF). University of Maryland Department of Computer Science. Archived (PDF) from the original on 2014-12-12. Retrieved June 6, 2020.
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
- ^ Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
- ^ Salomaa (1973), p. 59, Theorem 6.7
- ^ Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
- ^ Hopcroft & Ullman 1979, p. 205, Theorem 8.15.
- ^ Hopcroft & Ullman 1979, p. 206, Theorem 8.16.
- ^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
- ^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
- ^ John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003). Introduction to Automata Theory, Languages, and Computation. Addison Wesley. Here: Sect.7.6, p.304, and Sect.9.7, p.411
- ^ a b Yehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.
- ^ Hopcroft & Ullman 1979.
- ^ "How to prove that a language is not context-free?".
Works cited
[edit]- Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN 9780201029888.
- Salomaa, Arto (1973). Formal Languages. ACM Monograph Series.
Further reading
[edit]- Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free Languages and Push-Down Automata". In G. Rozenberg; A. Salomaa (eds.). Handbook of Formal Languages (PDF). Vol. 1. Springer-Verlag. pp. 111–174. Archived (PDF) from the original on 2011-05-16.
- Ginsburg, Seymour (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill.
- Sipser, Michael (1997). "2: Context-Free Languages". Introduction to the Theory of Computation. PWS Publishing. pp. 91–122. ISBN 0-534-94728-X.